Question

The hypotenuse of a triangle measures 12 inches, and the measure of one of the acute angles is two less than three times the measure of the other acute angle. What is the area of the triangle? Round your answer to the nearest square inch. (Note: The area of a triangle is 1/2bh, where the base and the height of the triangle are perpendicular).

Answer 1

26 sq in.

Explanation:

Let x = measure of one acute angle

3x - 2 = measure of other acute angle

Since the triangle is a right triangle, the sum of the measures of the acute angles is 90

x + 3x - 2 = 90

4x - 2 = 90

4x = 92

x = 23

90 - 23 = 67

The acute angles are 23° and 67°

Now, let a = length of side opposite the 23° angle

sin 23 = a/12

a = 12 sin 23

Let b = length of the side opposite the 67° angle

sin 67 = b/12

b = 12 sin 67

Area = 1/2ab = 1/2(12 sin 23°)(12 sin 67°)

= 25.90

26 sq in to the nearest inch

Wow! That is quite the problem.